A tale of two Bethe ans\"atze

TL;DR

This paper compares two Bethe ansatz methods for the Zamolodchikov-Fateev model, deriving explicit relations between eigenvectors and providing a scalar product formula, enhancing understanding of integrable models.

Contribution

It introduces a simple relation between eigenvectors from two Bethe ansatz constructions and derives the Slavnov scalar product formula for Tarasov-Bethe vectors.

Findings

Explicit relation between two Bethe ansatz eigenvectors.

Derivation of the Slavnov scalar product formula.

Enhanced understanding of eigenvector structures in the model.

Abstract

We revisit the construction of the eigenvectors of the single and double-row transfer matrices associated with the Zamolodchikov-Fateev model, within the algebraic Bethe ansatz method. The left and right eigenvectors are constructed using two different methods: the fusion technique and Tarasov's construction. A simple explicit relation between the eigenvectors from the two Bethe ans\"atze is obtained. As a consequence, we obtain the Slavnov formula for the scalar product between on-shell and off-shell Tarasov-Bethe vectors.